On Markovian semigroups of Lévy driven SDEs, symbols and pseudo-differential operators

Abstract

We analyse analytic properties of nonlocal transition semigroups associated with a class of stochastic differential equations (SDEs) in ${\mathbb{R}}^d$ driven by pure jump-type Lévy processes. First, we will show under which conditions the semigroup will be analytic on the Besov space $B_{p,q}^ m({\mathbb{R}}^d)$ with $1\le p, q\lt\infty$ and $m\in{\mathbb{R}}$. Secondly, we present some applications by proving the strong Feller property and give weak error estimates for approximating schemes of the SDEs over the Besov space $B_{\infty,\infty}^ m({\mathbb{R}}^d)$. The choice of Besov spaces is twofold. First, observe that Besov spaces can be defined via the Fourier transform and the partition of unity. Secondly, the space of continuous functions can be characterised by Besov spaces.